A generalized segregation model for concurrent dendritic, peritectic and eutectic solidification

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Acta Materialia 57 (2009) 2066–2079 www.elsevier.com/locate/actamat

A generalized segregation model for concurrent dendritic, peritectic and eutectic solidification D. Tourret, Ch.-A. Gandin * MINES ParisTech, CEMEF UMR CNRS 7635, 06904 Sophia Antipolis, France Received 10 December 2008; received in revised form 9 January 2009; accepted 9 January 2009 Available online 21 February 2009

Abstract A microsegregation model for the solidification of binary alloys is presented. It accounts for diffusion in all phases, as well as nucleation undercooling and growth kinetics of the solidifying microstructures. It successively considers the occurrence of several phase transformations, including one or several peritectic reactions and one eutectic reaction. Volume-averaged mass conservation equations are coupled with a heat balance assuming a uniform temperature. The model is applied to simulate the solidification of Al–Ni atomized droplets. Coupled with an atomization model, it predicts recalescences, volume fractions of phases as well as average compositions during solidification. It eventually predicts the occurrence of a recalescence during the growth of each microstructure, and the progress of peritectic transformations consuming previously formed solid phases. The influence of several process parameters is evaluated. A comparison with experimental measurements is given, showing variation of phase fractions as a function of particle size for Al–Ni atomized droplets. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Solidification; Segregation; Modeling; Peritectic; Atomization

1. Introduction Raney Ni alloys are used as powerful catalysts for hydrogenation reaction, such as in fuel cells [1]. These catalysts are produced from fine Al–Ni powders, usually produced by the gas atomization process. The catalytic performance of atomized Raney Ni alloys has been shown to be dependent on the fraction of phases in the solidified powders [2]. Moreover, a strong dependence has been demonstrated between processing parameters, such as droplet size, and the fraction of phases obtained after atomization [3,4]. A key issue concerning Raney Ni alloys is the occurrence of several phase transformations (dendritic, peritectic, and eutectic) taking place in the presence of liquid. As this is the case for most of the common metallic alloys, a study has been undertaken to build a multiple phase transformation microsegregation analysis. *

Corresponding author. E-mail address: [email protected] (Ch.-A. Gandin).

The peritectic solidification models found in the literature are either oversimplified [5] or based on a numerical solution of a multiphase diffusion problem with some front tracking method of the interfaces between phases [6,7]. These models do not consider phase transformations taking place prior to or after the peritectic reaction. Heringer et al. proposed that predictions of the solidification of atomized droplets may be obtained by considering a single equiaxed crystal nucleation event occurring at the center of a spherical domain [8]. This approach has recently been validated with experimental comparisons for the solidification of electromagnetic levitated droplets of Al–Cu alloys [9]. This contribution presents a semi-analytical multiple phase transformation model based on a volume-averaging method [9,10]. It considers the effect of diffusion in each phase, as well as nucleation undercooling and growth kinetics of each microstructure. An application to equiaxed solidification of Al–Ni atomized droplets is given, with a comparison with experimental measurements.

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.01.002

D. Tourret, Ch.-A. Gandin / Acta Materialia 57 (2009) 2066–2079

2. Modeling The model is based on an extension of the segregation analysis proposed by Beckermann and coworkers for equiaxed solidification [10,11], which considers the formation of a primary dendritic structure into an undercooled liquid. It has then been extended to consider the formation of a eutectic structure growing into the interdendritic liquid and then in the bulk undercooled liquid up to the completion of solidification [9]. In this work, the same approach is used to include peritectic transformations. 2.1. Mass conservation equations In a multiphase binary system with no velocity fields of the phases and no mass exchange with its surrounding environment, the conservation of the total mass of an a-phase and the conservation of the mass fraction of a solute species in the a-phase, wa, are respectively written: @qa ¼0 @t @ðqa wa Þ þ div j a ¼ 0; @t

ð1Þ ð2Þ

where qa is the density of the a-phase and ja is the diffusive flux of the solute species in the a-phase. The corresponding mass balances at an a/b interface are written: qa va=b  na=b þ qb va=b  nb=a ¼ 0

ð3Þ

½qa wa va=b  j a   na=b þ ½qb wb va=b  j b   nb=a ¼ 0;

ð4Þ

a/b

b/a

where n =  n is the normal unit vector of the a/binterface, directed from the a-phase toward the b-phase, and va/b is the velocity vector of the a/b-interface. 2.2. Averaging theorems The averaging operator of a quantity w, in the a-phase over the volume V, hwai, is defined using the phase presence function, ca, equal to 1 in the a-phase and 0 elsewhere: Z 1 ca wa dV : ð5Þ h wa i ¼ V V Similarly, the intrinsic averaging operator is defined with respect to the volume Va of the a-phase: Z 1 a a ca wa dV ¼ hwa i=ga ð6Þ hw i ¼ a V Va

Similar expressions can be derived for the time derivative of a vector, howa/oti, or the gradient of a scalar w, hgrad wai, which are not useful in the following developments. 2.3. Averaged mass conservation equations Applications of Definitions (5) and (6) and Theorems (7) and (8) to Eqs. (1)–(4) lead to the averaged equations: ! a=b X 1 Z @hqa i a a=b q v  n dA ð9Þ ¼ @t V Aa=b bðb–aÞ ! a=b X 1 Z @hqa wa i a a a a=b þ divhj i ¼ q w v  n dA @t V Aa=b bðb–aÞ  X 1 Z þ j a  na=b dA ð10Þ V Aa=b bðb–aÞ Z Z 1 1 a a=b a=b q v  n dA þ qb va=b  nb=a dA ¼ 0 ð11Þ V Aa=b V Aa=b Z Z 1 1 qa wa va=b  na=b dA þ j a  na=b dA V Aa=b V Aa=b 1 þ V

Z

b

b a=b

q w v

n

Aa=b

b=a

1 dA þ V

Z

j b  nb=a dA ¼ 0; ð12Þ Aa=b

where the right-hand-side term of Eq. (9) defines the exchange rate of the total mass of the a-phase due to the velocity of the a/b-interface, the first right-hand-side term of Eq. (10) defines the exchange rate of the mass of solute in the a-phase also due to the velocity of the a/b-interface and the second right-hand-side term of Eq. (10) stands for the exchange rate of the mass of solute in the a-phase due to diffusion. For the interfacial transfer terms, the following simplifications are applied: Z 1 a=b qa va=b  na=b dA ¼ S a=b qa vn ð13Þ V Aa=b Z  1 Da  a j a  na=b dA ¼ Sa=b a=b qa wa=b  hqa wa i ð14Þ V Aa=b l Z 1 a=b qa wa va=b  na=b dA ¼ Sa=b qa wa=b vn ; ð15Þ V Aa=b where qa and wa=b are averaged quantities in the a-phase a=b

ð7Þ

at the a/b-interface over surface Aa/b and vn stands for the average normal velocity of the a/b-interface; Da denotes the diffusion coefficient of solute in the a-phase and Sa/b = Aa/b/V (=Sb/a) stands for the interfacial area concentration. The length, la/b, is introduced in Eq. (14) to characterize diffusion of the species in the a-phase from the a/b-interface. Its evaluation is based on geometrical considerations and boundary conditions, and is defined as:

ð8Þ

la=b ¼

where ga stands for the volume fraction of the a-phase. The above definitions apply to both scalar or vector quantities. Averaging theorems for the derivative over time and space of the mean are written: !  Z a=b @wa @hwa i X 1 a a=b w v  n dA ¼  @t @t V Aa=b bðb–aÞ  X 1 Z wa  na=b dA : hdivwa i ¼ divhwa i þ V Aa=b bðb–aÞ

2067



wa=b  hwa ia  : a qa @n@wa=b a=b

ð16Þ

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D. Tourret, Ch.-A. Gandin / Acta Materialia 57 (2009) 2066–2079

2.4. Geometry The system chosen is a spherical symmetry one dimensional domain of radius R, with an external surface A and a volume V.

black in Tables 1 and 2 correspond to the step of the solidification sequence shown in Fig. 1. Additional expressions in grey correspond to possible interfaces that may form earlier or later during solidification. For instance, interface ð3Þ ð3Þ s1 =s3 may exist if no peritectic phase forms or remains ð3Þ

2.5. Zones and sequences For each new phase nucleated, a new zone is created at the center of the spherical domain. Its radial propagation is considered as well as mass exchanges between phases within each zone and with neighboring zones. After the nucleation of the first solid structure, i.e. the primary dendritic structure, the liquid phase is split into two liquid ‘‘phases”, each corresponding to different diffusion length scales, as was previously proposed [9,10]. The interdendritic liquid and the primary solid structure included in mushy ð1Þ zone (1) are denoted l(1) and s1 , respectively. The bulk liquid outside the mushy zone is denoted l(0). The solid ð1Þ s1 and the liquid l(0) are only in pointwise contact, and ð1Þ

ð1Þ

ð0Þ

the interfacial area between s1 and lð0Þ ; As1 =l , is ð1Þ neglected. Then, mass exchanges take place at the s1 =lð1Þ interface and at the lð0Þ =lð1Þ boundary. When a second nucleation event takes place, a second mushy zone extends from the center of the domain. This mushy zone (2) is comð2Þ posed of the newly formed secondary solid phase, s2 , the ð2Þ primary solid s1 , and the liquid l(2). The solidification sequence goes on until the occurrence of the nucleation of a eutectic structure, sðnÞ n , and its successive growth in the remaining liquids l(n1), l(n2), . . ., l(2), l(1), l(0). Fig. 1 illustrates the concurrent propagation of a primary dendritic microstructure s1, one secondary peritectic microstructure s2 and a eutectic microstructure s3. The schematic is thus limited to one step of a solidification sequence in which three zones successively form. Positions of the growing fronts of the microstructures are defined by ðmÞ

RðmÞ ð¼ R sm =1 , a simplified notation being introduced) with m 2 [1, 3]. The presentation of the model given below is heavily based on the schematics proposed in Fig. 1.

ðgs2 ¼ 0Þ. The eutectic structure also successively propagates in mushy zone (2) (up to R(3) > R(2)) and in mushy zone (1) (up to R(3) > R(1)). The interfacial area concentrations of the boundaries delimited by the same phase in two contiguous zones are evaluated with the radial position R(m) and the fraction ðmÞ

ðmÞ

the diffusion length in liquid lðmÞ ; lðmÞ=ðnÞ ð¼ l1 fied notation being introduced) [11]:

2.6. Mass transfers ðmÞ=ðnÞ

The external environment does not exchange mass with the system at the limit of the domain. The macroscopic solute diffusion terms entering the left-hand-side of Eq. (10) are neglected because their magnitude is much lower than their microscopic counterpart due to diffusive fluxes at the interfaces between phases, i.e. the last term on the right-hand-side of Eq. (10). Fig. 1a gives an illustration of the existing phase interfaces and zone boundaries considered for mass exchanges at a given step in a solidification sequence with concurrent dendritic, peritectic and eutectic reactions. Only a few interfaces and boundaries are active for mass transfers in Fig. 1. The expressions derived for their interfacial area concentration and diffusion lengths are available in Tables 1 and 2. Only the expressions in

ðmÞ

of phases in each zone, gsi and gl ð½i; m 2 ½1; 3Þ. For interfaces inner to a zone, a plate-like geometry that represents the secondary dendrite spacing created by primary solidification is assumed [10]. Simplification of notations in the tables is possible by defining the volume fraction P ðmÞ ðmÞ of the zone (m): gðmÞ ¼ g1 þ i gsi . Expressions for the interfacial area concentrations are summarized in Table 1. The expressions of the diffusion lengths are derived from Eq. (16) introducing assumed composition profiles in the different phases and in the different zones. Schematics are drawn in the radial and non-radial directions in Fig. 1b and c, respectively. In the radial direction, a zone (m) is delimited by radii R(n) and R(m) (e.g. zone (2) is delimited by R(3) and R(2)). The composition profiles in its solid phases are approximated using quadratic parabola with a no-flux condition ð2Þ when reaching the limit of the solid phase (e.g. s2 as illustrated in Fig. 1a and b). If the solid phase is present in the two neighboring zones on each side of zone (m), the profiles are still approximated using quadratic parabola with ð2Þ a no flux condition at the center of zone (m) (e.g. s1 is surð1Þ ð3Þ rounded by s1 and s1 ). In the liquid l(m), a quasi-stationary diffusion profile is assumed in the radial direction [10]. Taking as boundary conditions the composition at the moving boundary l(m)/l( n) and the average composition in the liquid l(m), the following expression is derived for

l

=lðnÞ

, a simpli-

  ðmÞ  ðnÞ R ¼ Pe 1 3 3 exp RðnÞ RðmÞ  RðnÞ ! 2 2 RðnÞ RðmÞ RðnÞ   RðmÞ 2 þ ðnÞ ðnÞ Pe Pe ! 2 2 3 RðnÞ RðnÞ RðmÞ PeðnÞ  þ ðnÞ 2 þ 1  2 R PeðnÞ PeðnÞ RðnÞ !! RðmÞ ðmÞ3 ðnÞ R exp Pe 1 RðnÞ !# IvðPeðnÞ RðmÞ =RðnÞ IvðPeðnÞ Þ   ; PeðnÞ RðmÞ =RðnÞ PeðnÞ RðnÞ

2

ð17Þ

D. Tourret, Ch.-A. Gandin / Acta Materialia 57 (2009) 2066–2079

a

R(3)

0

R(2)

(3)

vn

s(2) 2

(1) (1)

vsn1

/l

(2) (2)

vsn2 s(3) 3

/l

l(2)

l(1)

v(3) n

b

(0)

s(1) 1

(2) (2)

vsn2 /s1

(3) (3) s1 /s2

R(0)=R

(1)

s(2) 1

s(3) 2

R(1)

(2)

s(3) 1

2069

l(0)

v(2) n

v(1) n

w (2)

〈wl 〉l

wl

(2)

l(1)/(0)

s3

w

(1)

(3)

ws2

wl

(3)

〈ws2 〉s2

〈wl 〉l

(2) (1)

/l

(0)

(2)

(3) (2) /s2

/l

〈w 〉

(0)

(2)

〈ws2 〉s2

ws2 ws1

〈wl 〉l

(1) (0)

wl

(3) s1

(1)

l(0)/(1)

(3) s1 (2)

(2)

〈ws1 〉s1

(3) (2) s1 /s1

(1)

(1)

〈ws1 〉s1

w

(2) (1) /s1

ws1

c

r 0

(3)

ws1 s(3) 1 s(3) 1

g λ2 g(3) 2 s(3) 2

w (3)

w

0

/s2

(3) (3)

s2 /s1

w

0

(2) (2)

s1 /s2

w

s(2) 1

(2) (2) /s1

ws2 s(2) 1

g λ2 g(2) 2 s(2) 2

s(1) 1 l(0) (1) (1) l /s1

w (2) (2)

/l

g λ2 g(1) 2 (1) (1)

l(2)

1-g λ2 g(3) 2

1-g λ2 g(2) 2 wl(2)/s(2) 2

s(3) 3

l(2)

λ2 2

λ2 2

Homogeneous

s(1) 1

ws2 s(3) 3

w

ws1

/l

l(1)

λ2 2

Fig. 1. One-dimensional schematics of the multiple phase transformation model showing (a) zones (m), m 2 [1, 3], with concurrent developments of a ðmÞ ðmÞ ðmÞ dendritic structure, s1 , a peritectic structure, s2 , and a eutectic structure, s3 , into liquids, l(m), interfaces between structures (plain lines) and boundaries ðmÞ=1 sm between zonesðmÞ(dashed lines). The average normal velocities and radii corresponding to the position of the growing microstructures are labeled vðmÞ n ¼ vn ðmÞ ðm1Þ ðmÞ sm =1 a=b and R ð¼ R Þ, respectively, with m 2 [1, 3]. Other velocities defined by interfaces between phases are labeled, vn with ½a; b 2 ½si ; l ; ½i; m 2 ½1; 3. The corresponding composition profiles (plain lines) and average compositions, hwi (dashed lines), in the structures and liquids, are shown along (b) the direction of the propagation of the structures and (c) the microstructure length scale chosen to account for diffusion in the solid, k2. For the sake of clarity, not all the diffusion lengths defined at interfaces between phases and at boundaries between zones are sketched. As an example, illustration of diffusion ð0Þ ð1Þ ð1Þ ð0Þ lengths lð0Þ=ð1Þ ð¼ ll =l Þ and lð1Þ=ð0Þ ð¼ ll =l Þ are sketched at the l(0)/l(1) boundary. The notation has been simplified compared to the core of the text by dropping the bars on average quantities such as velocities and interfacial compositions.

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D. Tourret, Ch.-A. Gandin / Acta Materialia 57 (2009) 2066–2079

Table 1 Interfacial area concentrations, Sa/b = Aa/b/V(=Sb/a), corresponding to (black) the step of the solidification sequence schematized in Fig. 1 and (grey) the potential former and subsequent steps to Fig. 1.

Table 2 Diffusion lengths, la/b, at the a/b boundaries or interfaces corresponding to (black) the step of the solidification sequence schematized in Fig. 1 and (grey) the potential former and subsequent steps to Fig. 1. Symbol ‘‘–” is used for a non-existing interface (i.e. Sa/b = 0). Symbol ‘‘ ”is used for an interface involving the eutectic structure, where no diffusion occurs.

D. Tourret, Ch.-A. Gandin / Acta Materialia 57 (2009) 2066–2079

where R(n) and R(m) are respectively the radii of the internal and external boundary defining zone (m) that contains li1 ´ clet number defined quid lðmÞ ; PeðnÞ ¼ RðnÞ vðnÞ n =D is the Pe (n) by the position R of the internal boundary, its velocity, ðnÞ

sn =1 vðnÞ , a simplified notation being introduced), the n ð¼ vn diffusion coefficient in the liquid, D1, and Iv(x) = x exp(x)E1(x) is the Ivantsov function, where E1(x) is the exponential integral function. While the above expression was only proposed for the extradendritic liquid (i.e. m = 0 and n = 1) [11], it is also used for any interdendritic liquid (e.g. m = 1 and n = 2 define l(1)/(2)). The present model assumes that the diffusion lengths on each side of the liquid/liquid boundaries are the same, e.g. l(1)/ (0) = l(0)/(1) at the boundary between the interdendritic liquid l(1) defined in mushy zone (1) and the extradendritic liquid l(0) defined in zone (0). In non-radial directions into a zone (m), diffusion ðmÞ lengths in phases si and/or l(m) are also derived assuming parabolic composition profiles [9,10]. For that purpose, the length scale used is based on the secondary dendrite arm spacing, k2 (Fig. 1c). No mass transfer is assumed between the eutectic structure and the other solid phases. The composition of the eutectic structure that forms is taken equal to the one of the liquid in which it is growing. Expressions for the diffusion lengths are summarized in Table 2.

2.7. Continuity In a single solid phase si (i 2 [1, 3]) and for the liquid phase l present in two contiguous mushy zones (m) and (n) ([m, n] 2 [1, 3]), the composition is continuous through its boundary: ðmÞ

wsi wl

ðmÞ

ðnÞ

ðnÞ

=si

¼ wsi

=lðnÞ

¼ wl

ðnÞ

ðmÞ

=si

ð18Þ

=lðmÞ

ð19Þ

2.8. Thermodynamic equilibrium

2071

2.9. Nucleation All nucleation events take place at the center of the domain. The nucleation of a solid structure si, i 2 [1, 3], occurs at a defined undercooling DT sNi below its equilibrium i , thus corresponding to a nucleation temtemperature T sEq si i perature T N ¼ T sEq  DT sNi . 2.10. Growth The structures propagate from the center of the domain in the radial direction. Their growth kinetics define the boundary velocity between zones. The nucleated dendritic ð1Þ structure s1 grows into the extradendritic liquid l(0). The radius of the mushy zone, R(1), is calculated by a time inteð1Þ

gration of the dendrite tip velocity, vn , deduced from the ð1Þ=lð0Þ

supersaturation, Xsl , via the Kurz et al. model [12]:   s1 =1 1 D mEq w1=s1  w s1 =1

2 ð1Þ vn ¼  Iv1 ðX s1 =1 Þ ð20aÞ s1 =1 2 pC h i.h i 1ð0Þ Xs1 =1 ¼ w1=s1  h w1ð0Þ i w1= s1  ws1 =1 ; ð20bÞ s =1

1 is the slope of the liquidus line defined by the where mEq equilibrium between phases s1 and 1; Cs1 =1 is the Gibbs– Thomson coefficient of the primary solid phase s1 in the liquid. Because the primary dendritic structure can only ð1Þ grow as s1 in the liquid 1(0), notations have been simplified in Eqs. (20a) and (20b) by using s1 and l. Peritectic growth is assumed to follow the same growth kinetics model provided by Eqs. (20a) and (20b). Using the s2 =1 , the supersaturacorresponding properties, Cs2 =1 and mEq ð2Þ s2 =1 , is set to depend on both the average composition X tions hw1(o)i1(o) and hw1(1)i1(1) as:

2 6 Xs2 =1 ¼ 4w1=s2 

ð0Þ

g1

D ð0Þ E1ð0Þ D ð1Þ E1ð1Þ 3, ð1Þ h i w1 þ g1 w1 7 w1=s2  ws2 =1 ; 5 ð0Þ ð1Þ g1 þ g 1

ð20cÞ

in order to permit the peritectic mushy zone (2) of radius R(2) to pass over the primary mushy zone (1) and continue ð2Þ

A simplified phase diagram is considered, with linear monovariant lines separating single-phase equilibrium domains from regions where a mixture of two different phases is in equilibrium. Equations of these monovariant lines link the temperature, T, to the mass fraction of solute, w, through a linear approximation. The interfaces between phases are supposed to follow thermodynamic equilibrium. Thus, compositions at interfaces between different phases, wsi =sj–i and wsi =1 ; ½i; j 2 ½1; 3, are known as a function of temperature. The equilibrium assumed at the solid/liquid interfaces with the continuity of composition at the boundaries directly implies finite diffusion in all liquid phases. Indeed, two contiguous liquids located in mushy zones, e.g. l(1) and l(2) in Fig. 1, cannot be homogeneous without a discontinuity of composition at the boundary l(1)/l(2).

its growth in the extradendritic liquid l(0). The velocity v n is then defined with the interfacial compositions taken at ð2Þ the s2 =1 interface using Eq. (20a). Globulitic growth may occur if solidification leads to a fraction of solid greater than the mushy zone fraction in ð1Þ which it develops, e.g. gs1 > gð1Þ . In such a case, the approach proposed by previous authors is applied [9–11]. The growth velocity of the mushy zone is adapted to find a solution while constraining the system to reach an internal solid fraction equal to 1 in the mushy zone. The velocity of the expanding mushy zone is then directly derived from the formation of the internal solid phases. Radial propagation also takes place for the eutectic ð3Þ

structure at velocity vn . The position of the interface between the eutectic and the liquid is deduced from the Jackson and Hunt kinetics model [13]:

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D. Tourret, Ch.-A. Gandin / Acta Materialia 57 (2009) 2066–2079

ð3Þ

vn ¼ DT 2 =ð4K r K c Þ ð3Þ

ð21aÞ

k2 vn ¼ K r = K c

ð21bÞ

kDT ¼ 2K r . 



 K c ¼ w0 = 2pD1 jmEq jjm0Eq j jmEq j þ jm0 Eq j h i. K r ¼ 2ð1  fÞjm0 Eq jC sinðhÞ þ 2fjm Eq jC0 sinðh0 Þ h  i fð1  fÞ j mEq j þ jm0Eq j ;

ð21cÞ ð21dÞ

ð21eÞ

where DT = TE  T is the undercooling of the eutectic structure, w0 is the difference in composition between the limits of the eutectic tieline, f is the fraction of the previously formed solid phase in the eutectic structure (i.e. s2 in Fig. 1), C and C0 are the Gibbs–Thomson coefficients defined with respect to the first and second eutectic solid phases, and h and h0 are the contact angle between the two solid phases and the liquid. 2.11. Properties The density, q, the enthalpy of fusion per unit volume, DHf, and the heat capacity per unit volume, cp, are constant. 2.12. Heat transfer The temperature of the domain is assumed uniform. Heat exchange occurs at the outer boundary of the domain with a heat extraction rate, q_ ext ¼ hext ðT  T ext Þ. A Fourier boundary condition is assumed, with a convective heat transfer coefficient, hext, and an external temperature, Text. A global heat balance can thus be derived: ! X @gsðmÞ i @T A ¼  hext ðT  T ext Þ þ DH f cp ð22Þ @t @t V i;m where ðmÞ T is the time-dependent temperature of the system, and gsi are the fractions of the solid structures si in zone (m). The external temperature, Text, is taken as constant over time, while a time-dependent convective heat transfer coefficient, hext, is evaluated during the solidification process, accounting for the droplet size and process parameters as described by Henein et al. [14,15]. A tracking procedure with time of the droplet velocity as it falls through the atomization chamber is applied by solving the force balance: dv q  qf q Cd 2 g  0:75 f v: ¼ q q D dt

ð23aÞ

The atomization fluid being assumed stagnant, v is the velocity of the droplet. qf is the density of the atomization gas, D is the droplet diameter and Cd is the drag coefficient calculated from [16]: 18:5 ð23bÞ C d  0:6 ; Re where Re = Dqfv/lr is the Reynolds number, with lr the atomizing gas viscosity. The convective heat transfer coef-

ficient, hext, is computed through the expression of the Nusselt number, Nu, determined from the modified Whitaker correlation [14] as: hext D jf   B T mþ1  T mþ1 ext ¼2 jext ðm þ 1Þ T  T ext

Nu ¼



þ 0:4Re

1=2

þ 0:06Re

2=3

Pr

0:4

l1 lp

!0:25 ;

ð23cÞ

where jf is the thermal conductivity of the atomizing gas evaluated at the temperature of the droplet, T. Pr = lf/ (qfaf) is the Prandtl number, with af the thermal diffusivity of the gas. Finally, l1 and lp are the viscosity of the fluid in the bulk and on the droplet surface, respectively, evaluated at the temperature of the bulk fluid, Text, and at the droplet temperature. Since Re is a function of the droplet velocity, once the velocity is known, the heat transfer coefficient is evaluated using Eq. (23). 2.13. Numerical solution In each zone, the unknowns are the volume fraction and the average composition of the phases respectively computed from the total mass (Eq. (9)) and solute mass (Eq. (10)) balances. The velocities of the interfaces are evaluated through the total mass (Eq. (11)) and solute mass (Eq. (12)) balances, as well as the interface compositions deduced from the thermodynamic equilibrium conditions. In addition, at each boundary between zones, the velocity and the compositions are deduced from the growth kinetics models (Eq. (20) or (21)) and the continuity condition (Eq. (19)) and solute mass balance (Eq. (12)). The temperature of the system is calculated through the energy balance (Eq. (22)). The resulting system of partial differential equations is solved through an iterative Gear method. 3. Results 3.1. Application to Al–Ni alloys The model is applied to simulate the solidification of undercooled Al–Ni droplets produced by gas atomization. For the Raney Ni alloys with composition greater than 31.5 wt.% Al (50 at.% Al), a simplified equilibrium phase diagram using linear variations of the temperature with composition is a good approximation. Such a phase diagram with its metastable extensions using dashed lines is shown in Fig. 2. The main observation concerning this simplified diagram is the introduction of a domain of variation of the composition for the Al3Ni phase. This intermetallic phase is reported to have an exact stoichiometric composition in the literature [17–20]. However, if the mass fraction of solute in the Al3Ni phase is maintained constant, no composition gradient appears in this phase. According to

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Fig. 2. Linear approximation of the Al–Ni equilibrium phase diagram.

the equations of the model, no diffusion would then take place within the Al3Ni phase, which thus behaves as a diffusion barrier that prevents the peritectic reaction from taking place. The introduction of a composition range for the intermetallic phase should be seen as a numerical requirement that, as can be verified, has only minor consequences on the final simulation results. Similarly, as a consequence of the linearization of the phase diagram, the mass fractions of solute in the metastable phases could become greater than 100% at low temperature. This is obviously not satisfying and all equilibrium compositions are bounded so as to remain within 0 and 100%. This linear phase diagram is used as an input to the simulations presented below, in order to compute the equilibrium interfacial compositions as a function of temperature and the slope of the liquidus lines entering the growth kinetics models. In the calculations presented hereafter, the model is first run to simulate the solidification upon atomization of the Ni–50 wt.% Al (Ni–68.5 at.% Al) alloy. According to the simplified phase diagram presented in Fig. 2, primary solidification of the AlNi dendritic phase is expected below the liquidus temperature of 1619 K (1 ? s1). It is followed by the first peritectic reaction AlNi + l ? Al3Ni2 (s1 + l ? s2) below 1400.7 K. The second peritectic reaction Al3Ni2 + l ? Al3Ni (s2 + l ? s3) occurs below 1123.5 K up to the completion of solidification. However, in the case when full equilibrium is not achieved, some liquid could undergo the final eutectic transformation l ? Al3Ni + Al (l ? s3 + s4) below 914.8 K. 3.2. Limiting cases Preliminary simulations are presented to compare predictions, on the one hand with the lever rule approximation, i.e. assuming complete thermodynamic equilibrium of the system at any temperature, and on the other hand

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with the Gulliver–Scheil approximation, i.e. considering infinite diffusion in the liquid phase and no diffusion in the solid phases. For the primary dendritic structure and the eutectic structure, the Kurz et al. model [12] and the Jackson and Hunt model [13] are used. In this case, growth kinetics delay the transformations and consequently cause the simulation results to deviate from the exact lever rule and Gulliver–Scheil approximations. However, the peritectic reactions are assumed instantaneous. This means that at the nucleation temperature of a peritectic solid phase, the radius of the new zone is directly set equal to the radius of the mushy zone in which it forms. The peritectic solid phase then grows from a thin layer between the previously formed solid and the interdendritic liquid. In order to achieve better comparisons, the nucleation undercooling of the solid structures is set to zero and the enthalpy of fusion per unit volume is set to zero, permitting a better control of the cooling rate through a selection of the value of a constant heat transfer coefficient and a droplet radius. The value of the Fourier number in the solid phase, the product of the solidification time multiplied by the diffusion coefficient divided by the square of the diffusion length (i.e. the secondary dendrite arm spacing), characterizes the role of diffusion in the solid phase. Table 3 summarizes the values used in the simulations conducted for comparisons with the lever rule and Gulliver–Scheil approximations. In the case of the simulation to be compared with the lever rule approximation, a very slow cooling rate is imposed by selecting a large droplet radius, 100 lm, with a low heat transfer coefficient, 0.01 W m2 K1, as well as a large value of the diffusion coefficient in the solid phase, 1011 m2 s1. By comparison, in order for the simulation to be compared with the Gulliver–Scheil approximation, a much faster cooling rate is selected through a smaller droplet radius, 10 lm, a higher heat transfer coefficient, 100 W m2 K1, and no diffusion in the solid phases. Dummy values are used for the parameters entering the growth kinetics of the eutectic structure [9]. The results of the simulations, compared with the lever rule and the Gulliver–Scheil predictions, are shown in Fig. 3. The simulation to compare the model prediction with the lever rule approximation leads to a solidification time of about 1750 s, which is sufficiently slow to approach equilibrium conditions. Indeed, such a solidification time corresponds to a large value of the Fourier number thanks to the chosen values for the diffusion coefficient in the solid phase and the size of the secondary dendrite arm spacing (Table 3). As shown in Fig. 3a, the phase fractions predicted by the model (filled symbols) are very close to the lever rule approximation (empty symbols). A small difference is still to be noted at the very beginning of the solidification due to the dendritic growth kinetics. At the first peritectic temperature (1400.7 K), the peritectic Al3Ni2 phase rapidly grows to the detriment of the AlNi phase, which is fully replaced. A small development in the Al3Ni2 phase into the liquid is also visible due to the sudden decrease of the liquid fraction. When the second peritectic temperature is

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Table 3 Value of the simulation parameters to compare the model with the lever rule (Fig. 3a) and Gulliver–Scheil (Fig. 3b) approximations. Name

Symbol

Fig. 3a

Fig. 3b

Unit

Initial alloy composition Droplet radius Heat transfer coefficient External temperature Solid phases diffusion coefficient Liquid phase diffusion coefficient Secondary dendrite arm spacing Enthalpy of fusion per unit volume Heat capacity per unit volume Gibbs–Thomson coefficient of the solid phases Nucleation undercooling of the solid structure Fraction of Al3Ni-phase in the eutectic structure Eutectic structure contact angle (Al3Ni/Liquid) Eutectic structure contact angle (Al/Liquid)

w0 R h Text Ds D1 k2 DHf cp Cs DT sN f h h0

50 100 0.01 293 1011 109 1 0 106 107 0 0.6 60 65

50 10 100 293 0 109 1 0 106 107 0 0.6 60 65

wt.% Al lm W m2 K1 K m2 s1 m2 s1 lm J m3 J m3 K1 mK K – ° °

Fig. 3. Predicted evolution of the fraction of phases, ga with a 2 [Liquid, AlNi, Al3Ni2, Al3Ni, Al], as a function of the temperature upon solidification of an Al–50 wt.% Ni alloy with (a) infinite diffusion and (b) no diffusion in the solid structures. Comparisons are shown with the corresponding (a) lever rule and (b) Gulliver–Scheil approximations.

reached (1123.5 K), some of the Al3Ni2 phase reacts with liquid to form the peritectic Al3Ni phase. The peritectic reaction Al3Ni2 + liquid ? Al3Ni is made possible because of the Al3Ni stability domain introduced in the equilibrium

diagram. Otherwise, no composition gradient may appear in the peritectic solid, and the diffusion fluxes in the phase would always be nil. Below this temperature, solidification is complete. However, the equations of the model are still solved and the solid/solid reaction between the Al3Ni2 and Al3Ni phases can be simulated, still in agreement with the solution provided by the lever rule approximation. The second simulation compares the model predictions with the Gulliver–Scheil approximation. The result is shown in Fig. 3b. Because the solidification time is only 0.03 s, and thanks to the selected values of the parameters (Table 3), the value of the Fourier number is very small and solid diffusion has no effect. However, as the solidification is much faster, the influence of growth kinetics models on primary dendritic and eutectic structures is more important than in the previous simulation compared to the lever rule approximation. Upon primary dendritic growth of the AlNi phase, the system reaches a temperature more than 100 K below the liquidus temperature before the mushy zone is fully developed. The effect on the system is shown by the difference between the curves drawn for the volume fraction of the AlNi and liquid phases. However, these curves quickly catch up with the phase fractions using the Gulliver–Scheil approximation. Upon peritectic reaction, because no diffusion in the solid phase is considered, the interfaces between the solid phases are blocked and the fractions of previously formed phases are conserved. Only the interfaces with liquid are thus active. The influence of the eutectic growth kinetics thus affects the final fraction of eutectic as well as the second peritectic Al3Ni phase as was already shown previously [9]. During the eutectic reaction time, the Al3Ni solid phase continues to form in the interdendritic liquid, thus explaining the difference between curves for the Al3Ni and the Al phases with respect to the Gulliver–Scheil simulation. 3.3. Illustrative tests Three simulations are run to simulate gas atomization of Al–Ni alloy droplets, in order to illustrate the influence of

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Table 4 Value of the simulation parameters for atomization boundary condition under argon. Name

Symbol

Figs. 4–6

Unit

Particle density Aluminum density [20] Nickel density [20] Argon density [14] Argon viscosity [14] Argon thermal conductivity [14] Prandtl number [14] Whitaker correlation coefficients [14]

qp qAl qNi qf lf jf Pr B m Text V0

qAl w0 + qNi(1  w0) 2700 8908 539.23 T1.0205 2.38  107 T0.7913 1.86  104 T0.7915 0.713 1.86  104 0.7915 293 1

kg m3 kg m3 kg m3 kg m3 Pa s W m2 K1 – – – K m s1

External temperature Droplet initial velocity

Table 5 Value of the simulation parameters to illustrate the influence of process parameters. Name Initial alloy composition Fig. 4a and b Fig. 4c Droplet radius Fig. 4a and c Fig. 4b Solid phases diffusion coefficient Liquid phase diffusion coefficient Secondary dendrite arm spacing Enthalpy of fusion per unit volume Heat capacity per unit volume Gibbs–Thomson coefficient of the solid phases Nucleation undercooling of the solid structure Fraction of Al3Ni-phase in the eutectic structure Eutectic structure contact angle (Al3Ni/Liquid) Eutectic structure contact angle (Al/Liquid)

process and material parameters on the solidification of a droplet. The atomization process is integrated by evaluating the heat transfer coefficient at the external boundary of a droplet using Eqs. (23a)–(23c). Realistic values of the parameters have been chosen as summarized in Tables 4 and 5. The atomizing gas is argon, the initial droplet velocity is assumed to be 1 m s1 and the external temperature is 293 K. All these values are based on the literature where comparisons with experimental observations were achieved [14,15]. The growth kinetics models are applied to the expansion of each solid structure (dendritic, peritectic(s), and eutectic). For thermodynamic and physicochemical properties of the alloy, realistic orders of magnitude have also been taken. The diffusion coefficients in the liquid and solid phases are taken as 108 and 1012 m2 s1, respectively. Once the droplet is completely solidified, it is assumed that no further diffusion occurs. This is required because the diffusion coefficients are taken as independent of temperature, leading to predicted solid-state transformations up to room temperature. One way to avoid these transformations is to use Arrhenius’ laws for the diffusion coefficients, though this approach was not chosen here since the objective is to give illustrative cases of the model. In the absence of data, no nucleation undercooling is

Symbol

Fig. 4

Unit

w0

50 58

wt% Al

R

50 100 1012 108 1.5 109 106 108 0 0.6 60 65

lm

Ds Dl k2 DHf cp Cs DT sN f h h0

m2 s1 m2 s1 lm J m3 J m3 K1 mK K – ° °

assumed for the formation of solid structures. Secondary dendrite arm spacing is taken as 1.5 lm, based on measurements on atomized alloys [15]. Three distinct simulations are carried out to study the effect of the atomized particle size and the alloy composition. A reference calculation is done with a droplet of radius 50 lm, and an initial alloy composition of 50 wt.% Al. The second simulation is run with the same parameters, except the radius of the droplet, which is taken as 100 lm. For the third simulation, the radius is 50 lm, the same as for the reference case, but the alloy composition is set to 58 wt.% Al. The simulated predictions of temperature, phase fractions and average composition of phases are plotted in Fig. 4a–c, respectively. Let us first turn our attention to Fig. 4a, which presents the results of the reference calculation. In this simulation, the evolution of the volume phase fractions clearly shows four solidification steps: (i) primary dendritic formation of AlNi; (ii) growth of peritectic Al3Ni2; (iii) second peritectic reaction forming Al3Ni; and (iv) final eutectic structure Al3Ni + Al leading to completion of solidification. The formation of the two peritectic structures, Al3Ni2 and Al3Ni, occur both by solidification in the liquid and by transformations AlNi ? Al3Ni2 and Al3Ni2 ? Al3Ni,

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Fig. 4. Predictions of (top) cooling curves, (center) volume fractions, ga with a 2 [Liquid, AlNi, Al3Ni2, Al3Ni, Al], and (bottom) average compositions of structures, hwaia with a 2 [Liquid, AlNi, Al3Ni2, Al3Ni, Al], upon solidification of Al  w0 Ni droplets of radius R by gas atomization with (a) w0 = 50 wt.% Al, R = 50 lm, (b) w0 = 50 wt.% Al, R = 100 lm, and (c) w0 = 58 wt.% Al, R = 50 lm. The equilibrium temperatures of the different microstructures are added as horizontal lines in the cooling curves.

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respectively. The decrease of the volume fraction of AlNi during the first peritectic reaction is due, in the very first moments, to the remelting of primary dendrites, and then, to the peritectic solid–solid reaction. This AlNi ? Al3Ni2 transformation is a lot more active than the Al3Ni2 ? Al3Ni transformation taking place during the second peritectic transformation. Indeed, almost no Al3Ni2 phase is consumed by the second peritectic transformation. The diffusion coefficients in all the solid phases are the same, and so the activity of the reaction may only be caused by the shape of the two peritectic reactions in the phase diagram. The equilibrium composition range of the Al3Ni2 solid is much bigger than the one for Al3Ni. The composition gradient within the peritectic phase is more important in the first peritectic reaction than in the second one, and the transformation is thus more active. That analysis had been provided in 1987 by St. John and Hogan, and gives a simple way of predicting the peritectic transformation rate based solely on looking at the shape of the phase diagram [21]. Comparison of the simulation results of Fig. 4a with b gives access to the influence of the size of the droplet, the alloy composition being kept constant. The same solidification sequence detailed above is retrieved. However, the cooling rate is lower for the biggest particle. For the 50lm radius droplet, the model predicts a solidification time of about 0.03 s. The 100-lm radius droplet solidifies in about 0.09 s. The cooling rate strongly influences the final phase fractions in the droplet. Thus, the first peritectic reaction is found to fully consume the AlNi phase, leading to a larger amount of the Al3Ni2 peritectic phase in the large droplet (Fig. 4b). Four recalescences can be distinguished on the predicted cooling curves in Fig. 4a and b. They correspond to the growth of the four solid structures. It is to be noted that the first peritectic reaction causes a smaller recalescence than the primary dendritic, or the second peritectic reaction. While the first three recalescences are only slightly influenced by the cooling rate, the plateau following the eutectic recalescence is longer for the 50-lm radius droplet, relative to its solidification time. This result is consistent since the length of the plateau is proportional to the final volume fraction of the eutectic microstructure, which is known to increase with cooling rate and alloy composition. The latter parameter is indeed accessible when comparing the predictions of Fig. 4a with those of Fig. 4c. The liquidus temperature of the Ni–58 wt.% Al alloy is lower than the 50 wt.% Al alloy (1383.6 K) and below the peritectic temperature where the reaction AlNi + l ? Al3Ni2 occurs. Consequently, primary solidification takes place with the formation of a dendritic Al3Ni2 phase as predicted in Fig. 4c. Only one peritectic reaction is taking place, Al3Ni2 + l ? Al3Ni, below 1123.5 K. The recalescence due to the dendritic solidification of Al3Ni2 is larger than the one observed in the other simulations, where Al3Ni2 forms from a peritectic reaction. Also, as the liquidus temperature is lower for this alloy, the primary

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dendritic solidification has less time to occur and more liquid remains when the peritectic temperature, 1123.5 K, is reached. Consequently, more peritectic Al3Ni phase and eutectic Al + Al3Ni structures form. For the three simulations, the average composition of each phase during the solidification sequence has been calculated, and is plotted in Fig. 4. The AlNi and Al3Ni2 phases, which have quite high composition equilibrium ranges, exhibit variations of up to 3 wt.% Al, which may greatly influence the behavior of the final material. During the entire solidification sequence, the Al3Ni has a composition very close to the exact stoichiometric proportion (58 wt.% Al), showing the little influence of the domain of variation of the composition for the Al3Ni phase introduced in the phase diagram. The final composition of the eutectic structure is always close to the equilibrium composition (94 wt.% Al). The high cooling rates involved in this process explain why we are so far from equilibrium. A high cooling rate is, in fact, one of the properties expected from atomization, in order to process solidified structures far from equilibrium conditions. These calculations are only qualitative, but they give an overview of the influence of the alloy and process parameters. For instance, it shows how the presence or absence of a phase may be directly depend on process parameters such as the droplet size and the composition. 3.4. Experimental comparison In the framework of the IMPRESS project, the fraction of the phases in atomized droplets has been measured. After being sieved in different size ranges, the particles have been analyzed by neutron diffraction [3,4]. The presence and weight fraction of each phase have been determined, in the light of previous studies on the Ni–Al system [22]. A significant variation in phase fractions in terms of droplet size has been noted for a typical composition of Ni– 75 at.% Al (Ni–58 wt.% Al). This alloy corresponds to the simulation presented in Fig. 4c, with formation of a primary dendritic solid, Al3Ni2, only one peritectic phase, Al3Ni, and finally a eutectic structure Al + Al3Ni. Simulations have been run to compare with these experimental results. Thermophysical properties, such as latent heat of fusion and heat capacity, are taken from experimental assessment in the literature [23]. No data is yet available concerning the diffusion coefficients of Al in the different phases, and none have been measured in the framework of the project; thus the values have been fitted to approximate the experimental plot. A simulation has been run at the center and at each bound of the particle size ranges. The parameters used in the simulations are summarized in Table 6. The computed cooling rates at the time of nucleation and the solidification times are plotted in Fig. 5. The trends are in agreement with expected variations. Predicted variations of the fraction of phases with the particle size are shown in Fig. 6.

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Table 6 Value of the simulation parameters to compare with experimental results of atomized Al–58 wt.% Ni alloy under argon. Name

Symbol

Figs. 5 and 6

Unit

Initial alloy composition Droplet radius Solid phases diffusion coefficient Al3Ni2 Al3Ni Liquid phase diffusion coefficient Secondary dendrite arm spacing Enthalpy of fusion per unit volume [23] Heat capacity per unit volume [23] Gibbs–Thomson coefficient of the solid phases Nucleation undercooling of the solid structure Fraction of Al3Ni-phase in the eutectic structure Eutectic structure contact angle (Al3Ni/Liquid) Eutectic structure contact angle (Al/Liquid)

w0 R

58 9.5, 19, 22.75, 26.5, 32, 37.5, 45.25, 53, 64, 75, 90.5, 106

wt.% Al lm

Ds 1 Ds 2 D1 k2 DHf cp Cs DT sN f h h0

1011 1012 2  108 1 2.4  109 4.25  106 107 0 0.6 60 65

m2 s1

Fig. 5. Computed cooling rate when the temperature reaches the liquidus of the alloy (plain line) and solidification time (dashed line) as a function of the droplet size for gas atomized Ni–58 wt.% Al droplets.

Fig. 6. Predicted (dashed lines) vs. measured (plain lines) phase fractions in gas atomized Ni–58 wt.% Al droplets as a function of the droplet size.

m2 s1 lm J m3 J m3 K1 mK K – ° °

When the droplet size increases, the amount of peritectic plus eutectic phase Al3Ni increases, while the amount of primary Al3Ni2 and eutectic Al decreases. Experimentally measured trends in phase fraction variations with particle size are obtained from simulations. Nevertheless, the predicted amount of Al phase is always lower than the measured one, while the amount of phase Al3Ni2 is greater than the experimental data, and the slope of the predicted Al3Ni2 and Al3Ni curves shown in Fig. 6 are higher than the measured data. The influence of the diffusion coefficients on the predictions is crucial. At this point, experimental data is sorely lacking. The introduction of experimentally assessed diffusion coefficients of Al in the different phases and their variation with temperature would have a great impact on these simulation results. Some other data are missing from the Al3Ni–Al eutectic system, even though their influence has been asssessed as being less important than the diffusion coefficients. The presence of the three phases, Al, Al3Ni and Al3Ni2, has been experimentally proved. Nevertheless, thermodynamic computations showed that a metastable eutectic Al + Al3Ni2 might form below 881 K. A further analysis of the eutectic lamellae might be useful and the introduction of metastable eutectic growth is conceivable. Other modifications may effect the simulation results, such as considering different latent heats of fusion of the solid phases, as well as the heat corresponding to each solid–solid transformation. Moreover, we are still making some assumptions about the experimental results, and how we compare these with simulation results. Indeed, the estimations are obtained by assessing the phase fractions of a size range after sieving; the prediction with which these are to be compared should be an average value between the two bounds of the size range. An even better way of making this comparison may be achieved by giving several predictions for a size range and calculating an average value, and possibly balancing this by the distribution of the droplet radii. The neutron diffraction results may also be interpreted several ways considering different structural arrangements of the phases, and that may change the estimated weight fraction of

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phases. Furthermore, for the smallest droplets (radius < 30 lm), the presence of additional phases, such as metastable Al9Ni2 and quasi-crystalline phases, is suspected. These phases are known to appear at high cooling rates and with high Al contents [24–29]. They are taken into account neither in the analyses of the experiments nor in the simulations. Accounting for these phases would have a significant effect for the finest particles. However, introduction of these phases in the simulations would require a thermodynamic description, which is not yet available. 4. Conclusions A semi-analytical model, based on the volume-averaging method, is proposed to simulate the solidification and segregation of binary alloys experiencing concurrent multiple phase transformations (dendritic, one or several peritectic, and eutectic). The model takes into account the diffusion in every phase and the growth kinetics of the different microstructures. It predicts the time evolution of temperature and the average volume fraction and composition of structures. The model is applied to the Al–Ni system. Under appropriate conditions, limiting case results are retrieved, such as lever rule (high diffusion coefficients in all phases and/or slow cooling) and Gulliver–Scheil approximations (no diffusion in solids and/or fast cooling). An atomization model is used for the heat exchange boundary condition in order to simulate the Al–Ni droplet atomization process. The influence of several parameters such as the initial composition and size of the droplet are evaluated. Finally, a comparison is given with experimental phase fractions of atomized powders obtained by neutron diffraction analyses [3,4,22]. The model shows reasonable agreement with the behavior of the system, and identifies the key thermodynamic and physicochemical data required. This model represents a simple and practical tool with which to study the solidification of alloys experiencing multiple phase transformations, which is actually the case for most common metallic alloys. It is detailed enough to take into account several phenomena occurring during the solidification (undercooling, growth kinetics, different diffusion length scales, etc.) while leading to quite efficient calculations. It is thus believed to provide a practical solution for application to metallic alloys. Many prospective developments may be foreseen. Coupling with equilibrium calculation software is currently being implemented. While such a coupling is in most cases not crucial for binary alloys, it is required to extend the approach to multicomponent alloys, which are more rele-

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vant for most industrial applications. The introduction of density variations may also be undertaken. Furthermore, a coupling of the microsegregation model with larger-scale simulation may be envisaged, through coupling with finiteelement methods. Acknowledgments This work has been partly funded by the EU-FP6 integrated project IMPRESS (Contract No. NMP3-CT-2004500635). Experimental data were provided by M. CalvoDahlborg, U. Dahlborg (University of Rouen, France) and G. Reinhart (ESA, Grenoble, France). References [1] Jarvis DJ, Voss D. Mat Sci Eng A 2005;413:538. [2] Devred F, Gieske1 A, Adkins N, Bakker JW, Nieuwenhuys BE. Appl Catal A 2008, in press. doi:10.1016/j.apcata.2008.12.039. [3] Bao CM, Dahlborg U, Adkins N, Calvo-Dahlborg M. Acta Mater, submitted for publication. [4] Calvo-Dahlborg M, Chambreland S, Bao CM, Quelennec X, Cadel E, Cuvilly F, et al. Ultramicroscopy, in press. doi:10.1016/ j.ultramic.2008.10.028. [5] Lopez HF. Acta Metall Mater 1991;39:1543. [6] Thuinet L, Combeau H. J Mater Sci 2004;39:7213. [7] Tiaden J. J Cryst Growth 1999;198/199:1275. [8] Heringer R, Gandin Ch-A, Lesoult G, Henein H. Acta Mater 2006;54:4427. [9] Gandin Ch-A, Mosbah S, Volkmann Th, Herlach D. Acta Mater 2008;3023:3035. [10] Wang CY, Beckermann C. Metall Trans A 1993;24:2787. [11] Martorano MA, Beckermann C, Gandin Ch-A. Metall Mater Trans A 2003;34:1657. [12] Kurz W, Giovanola B, Trivedi R. Acta Metall Mater 1986;34:823. [13] Jackson KA, Hunt JD. T Metall Soc AIME 1966;236:1129. [14] Wiskel JB, Henein H, Maire E. Can Metall Quart 2002;41:97. [15] Prasad A, Henein H, Mosbah S, Gandin Ch-A. ISIJ Int, submitted for publication. [16] Yule AJ, Dunkley JJ. Oxford series on advanced manufacturing. Oxford: Clarendon Press; 1994. [17] Massalski TB. Binary alloy phase diagrams, vol. 1. Materials Park, OH: ASM; 1986. [18] Du Y, Clavaguera N. J Alloy Compd 1996;237:20. [19] Ansara I, Dupin N, Lukas HL, Sundman B. J Alloy Compd 1997;247:20. [20] Smithells CJ. Smithells metals reference book. 5th ed. Bodmin: Butterworths; 1983. [21] St John DH, Hogan LM. Acta Metall Mater 1987;35:171. [22] Taylor A, Doyle NJ. J Appl Cryst 1972;5:201. [23] IMPRESS project internal communication, Deliverable D7-3; 2007. [24] Yamamoto A, Tsubakino H. Scripta Mater 1997;37:1721. [25] Blobaum KJ, Van Heerden D, Gavens AJ, Weihs TP. Acta Mater 2003;51:3871. [26] Pohla C, Ryder PL. Mat Sci Eng A 1991;134:947. [27] Pohla C, Ryder PL. Acta Metall Mater 1997;45:2155. [28] Grushko B, Velikanova TYa. Powder Metall Met C+ 2004;43:72. [29] Abe E, Tsai AP. J Alloy Compd 2002;342:96.